mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be...

areas of mathematics, a set in a topological vector space is called bounded or von Neumann bounded, if every neighborhood of the zero vector can be inflated...

The space M is called precompact or totally bounded if for every r > 0 there is a finite cover of M by open balls of radius r. Every totally bounded space...

metric space (M, d) is a bounded metric space (or d is a bounded metric) if M is bounded as a subset of itself. Total boundedness implies boundedness. For...

definition of boundedness can be weakened a bit; E {\displaystyle E} is bounded if and only if every countable subset of it is bounded. A set is bounded if and...

of the topology used, in a particular theory. Compactly embedded Totally bounded space page 12 of V. Khatskevich, D.Shoikhet, Differentiable Operators...

admits no finite subcover. It is complete but not totally bounded. For an ordered space (X, <) (i.e. a totally ordered set equipped with the order topology)...

n-space, every ball is bounded by a hypersphere. The ball is a bounded interval when n = 1, is a disk bounded by a circle when n = 2, and is bounded by...

the measure space ( X , A , μ ) {\displaystyle (X,{\mathcal {A}},\mu )} is called a finite measure space or a totally finite measure space. For any measurable...

resulting space, with the quotient topology, is totally disconnected. However, by considering the two copies of zero, one sees that the space is not totally separated...

vector space X {\displaystyle X} is said to have the Heine–Borel property (R.E. Edwards uses the term boundedly compact space) if each closed bounded set...

space is complete, though complete spaces need not be compact. In fact, a metric space is compact if and only if it is complete and totally bounded....

term chain is sometimes defined as a synonym of totally ordered set, but generally refers to a totally ordered subset of a given partially ordered set...

discrete uniform or metric space is complete. Combining the above two facts, every discrete uniform or metric space is totally bounded if and only if it is...

lies entirely in K, see also totally convex. Cotangent bundle Covariant derivative Cut locus Diameter of a metric space is the supremum of distances between...

competitors. SpaceX, NASA and United States Armed Forces have a symbiotic relationship, bounded together by governmental contracts. SpaceX was founded...

lower) bound is said to be bounded from above or majorized (respectively bounded from below or minorized) by that bound. The terms bounded above (bounded below)...

viewed as a metric space, its open balls are the open bounded intervals (c + r, c − r), and its closed balls are the closed bounded intervals [c + r, c − r]...

functions of bounded variation on a compact interval are exactly those f which can be written as a difference g − h, where both g and h are bounded monotone...

Schwartz spaces are topological vector spaces (TVS) whose neighborhoods of the origin have a property similar to the definition of totally bounded subsets...

complete and totally bounded. A Fréchet–Montel space is a Fréchet space that is also a Montel space. A separable Fréchet space is a Montel space if and only...

any totally ordered set. It is a natural generalization of the topology of the real numbers to arbitrary totally ordered sets. If X is a totally ordered...

convergent sequence {xn} is bounded, by the uniform boundedness principle. Conversely, every bounded sequence in a Hilbert space admits weakly convergent...

converges (i.e. has a limit point in X). Totally bounded (or Precompact). A uniform space X is totally bounded if for each entourage E ⊂ X × X there is...

Wijsman topologies on Cl(X) coincide if and only if (X, d) is a totally bounded space. Hausdorff distance Kuratowski convergence Vietoris topology Hemicontinuity...

topology of uniform convergence on totally bounded subsets (instead of bounded subsets) in the definition of dual space X ′ . {\displaystyle X^{\prime }...

applied to Banach spaces. Although boundedness is the same as continuity for linear maps between normed spaces, the term "bounded" is more commonly used...

anything or that space exists "outside" it. To any observer in the universe, it appears that all but the nearest galaxies (which are bound to each other...

topological space is compact metrizable if and only if it is a continuous image of a Cantor space. Let C(X) denote the space of all real-valued, bounded continuous...

functionals on a preordered vector space X {\displaystyle X} that map every order interval into a bounded set is called the order bound dual of X {\displaystyle...